<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c4">X1</font>, <font color="Maroon" title="c5">X2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7_1_2"/><i><font color="Green" title="E3">A8</font></i>: 
 ex <font color="Olive" title="b1">p1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp;  ex <font color="Olive" title="b2">q1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">p1</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b2">q1</font> holds <br/><font color="Olive" title="b2">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c2">x</font> ) &amp; <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b2">q1</font> ) )
 <b>and </b><br/><a NAME="E2:7_1_2"/><i><font color="Green" title="E4">A9</font></i>: 
 ex <font color="Olive" title="b1">p2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp;  ex <font color="Olive" title="b2">q2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b2">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">p2</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b2">q2</font> holds <br/><font color="Olive" title="b2">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">x</font> ) &amp; <font color="Maroon" title="c5">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b2">q2</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">p1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:7_1_2"/><i><font color="Green" title="E5">A10</font></i>: 
<font color="Maroon" title="c6">p1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:7_1_2"/><i><font color="Green" title="E6">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E6:7_1_2"/><i><font color="Green" title="E7">A12</font></i>: 
 ex <font color="Olive" title="b1">q1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">q1</font> holds <br/><font color="Olive" title="b1">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c2">x</font> ) &amp; <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b1">q1</font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:7_1_2"><i><font color="Green" title="E3">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">p2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:7_1_2"/><i><font color="Green" title="E8">A13</font></i>: 
<font color="Maroon" title="c7">p2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E9:7_1_2"/><i><font color="Green" title="E9">A14</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E10:7_1_2"/><i><font color="Green" title="E10">A15</font></i>: 
 ex <font color="Olive" title="b1">q2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">p2</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">q2</font> holds <br/><font color="Olive" title="b1">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">x</font> ) &amp; <font color="Maroon" title="c5">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b1">q2</font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E2:7_1_2"><i><font color="Green" title="E4">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">q1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E12:7_1_2"/><i><font color="Green" title="E11">A16</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font>
 <b>and </b><br/><a NAME="E13:7_1_2"/><i><font color="Green" title="E12">A17</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">q1</font> holds <br/><font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E14:7_1_2"/><i><font color="Green" title="E13">A18</font></i>: 
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c8">q1</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E6:7_1_2"><i><font color="Green" title="E7">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">q2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E16:7_1_2"/><i><font color="Green" title="E14">A19</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">p2</font>
 <b>and </b><br/><a NAME="E17:7_1_2"/><i><font color="Green" title="E15">A20</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> holds <br/><font color="Maroon" title="c9">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E18:7_1_2"/><i><font color="Green" title="E16">A21</font></i>: 
<font color="Maroon" title="c5">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c9">q2</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E10:7_1_2"><i><font color="Green" title="E10">A15</font></i></a></i>;<br/>
<a NAME="E19:7_1_2"/><i><font color="Green" title="E17">A22</font></i>: 
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">p2</font> &amp;  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font> st <br/>( <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font> ) )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:7_1_2"><i><font color="Green" title="E5">A10</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2"><i><font color="Green" title="E6">A11</font></i></a>, <a class="txt" href="bhsp_5.html#E8:7_1_2"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E9:7_1_2"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font><b> such that </b><br/><a NAME="E21:7_1_2"/><i><font color="Green" title="E18">A23</font></i>: 
<font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font>
 <b>and </b><br/><a NAME="E22:7_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font>
 <b>and </b><br/><a NAME="E23:7_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E4:7_1_2"><i><font color="Green" title="E5">A10</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2"><i><font color="Green" title="E6">A11</font></i></a>, <a class="txt" href="bhsp_5.html#E8:7_1_2"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E9:7_1_2"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></i>;<br/>
<a NAME="E24:7_1_2"/><i><font color="Green" title="E19">A24</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">q1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">q1</font> )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E12:7_1_2"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E25:7_1_2"/><i><font color="Green" title="E20">A25</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E16:7_1_2"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="bhsp_5.html#E19:7_1_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E26:7_1_2"/><i><font color="Green" title="E21">A26</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font>
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:7_1_2"><i><font color="Green" title="E5">A10</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2"><i><font color="Green" title="E6">A11</font></i></a>, <a class="txt" href="bhsp_5.html#E8:7_1_2"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E9:7_1_2"><i><font color="Green" title="E9">A14</font></i></a>, <a class="txt" href="bhsp_5.html#E16:7_1_2"><i><font color="Green" title="E14">A19</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:7_1_2_1"/>
( <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">P</font> implies <font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">q1</font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E24:7_1_2"><i><font color="Green" title="E19">A24</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E2:7_1_2_1"/>
( <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> iff <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E25:7_1_2"><i><font color="Green" title="E20">A25</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:7_1_2"/><b>then </b><i><font color="Green" title="E22">A27</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E29:7_1_2"/>
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_2_2"/><i><font color="Green" title="E23">A28</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_2_2_1"/><i><font color="Green" title="E24">A29</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="bhsp_5.html#E1:7_1_2_2"><i><font color="Green" title="E23">A28</font></i></a></i>;<br/><a NAME="E3:7_1_2_2_1"/>
<font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">q2</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E25:7_1_2"><i><font color="Green" title="E20">A25</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green" title="E24">A29</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="bhsp_5.html#E1:7_1_2_2"><i><font color="Green" title="E23">A28</font></i></a></i>;<br/><a NAME="E5:7_1_2_2_1"/><i><font color="Green" title="E25">A30</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">q1</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E12:7_1_2"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="bhsp_5.html#E25:7_1_2"><i><font color="Green" title="E20">A25</font></i></a>, <a class="txt" href="bhsp_5.html#E26:7_1_2"><i><font color="Green" title="E21">A26</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green" title="E24">A29</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E6:7_1_2_2_1"/><i><font color="Green" title="E26">A31</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">P</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E16:7_1_2"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="bhsp_5.html#E19:7_1_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green" title="E24">A29</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><font color="Maroon" title="c9">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> = 
 <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E17:7_1_2"><i><font color="Green" title="E15">A20</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green" title="E24">A29</font></i></a></i>
<br/>.= 
 <a href="bhsp_5.html#K3" title="BHSP_5:func.3">PO</a> <font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E21:7_1_2"><i><font color="Green" title="E18">A23</font></i></a>, <a class="txt" href="bhsp_5.html#E6:7_1_2_2_1"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E13:7_1_2"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2_2_1"><i><font color="Green" title="E25">A30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E6:7_1_2_2_1"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E8:7_1_2_2_1"/>
<font color="Maroon" title="c9">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_2_2"/>
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E28:7_1_2"><i><font color="Green" title="E22">A27</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E30:7_1_2"/>
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X2</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E12:7_1_2"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="bhsp_5.html#E14:7_1_2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="bhsp_5.html#E18:7_1_2"><i><font color="Green" title="E16">A21</font></i></a>, <a class="ref" href="finsop_1.html#T8" title="FINSOP_1:th.8">FINSOP_1:8</a></i>;<br/>


</div>
